Rheology of Aqueous Foams: a Literature Review of Some

Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5, pp. 587-596
Copyright © 1999, Éditions Technip
Rheology of Aqueous Foams: a Literature Review
of some Experimental Works
B. Herzhaft1
1 Institut français du pétrole, 1 et 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - France
e-mail: [email protected]
Résumé — Rhéologie des mousses aqueuses : revue bibliographique de quelques travaux
expérimentaux — La mousse étant un système dispersé et instable par nature, sa caractérisation
rhéologique est délicate. De nombreux paramètres doivent être pris en compte et contrôlés : la qualité de
la mousse (fraction volumique en gaz), sa texture (distribution en tailles de bulles), la taille de l'appareil
de mesure par rapport à la taille des bulles, l'influence du mode de formation, le glissement aux parois et
la compressibilité de la mousse. D'autre part, la mousse doit être stable et ne pas évoluer dans les
conditions de la mesure. Ces nombreux paramètres expliquent qu'il n'y ait pas, à l'heure actuelle, de
consensus concernant le comportement de ce type de système. L'influence de la pression et de la
température (ne serait-ce que sur le comportement en statique de la mousse) est très peu étudiée. Une
étude expérimentale rigoureuse doit prendre en compte et contrôler tous les paramètres influençant la
stabilité et la structure de la mousse.
Mots-clés : rhéologie, mousses.
Abstract — Rheology of Aqueous Foams: a Literature Review of some Experimental Works —
Foam is a dispersed system and is unstable by nature, its rheological characterization is very
difficult. Numerous parameters have to be considered and controlled: foam quality, i.e. gas volume
fraction, foam texture (bubbles size distribution), size of the measurement apparatus compared to
bubbles size, influence of foam production method, wall slip phenomena and foam compressibility.
Foam must be stable and must not evolve during measurement time. These numerous parameters do
explain there is no general view concerning the behavior of this kind of system. Influence of pressure
and temperature has not been the subject of many studies, even on static foam. A rigorous
experimental method should consider and control each of these parameters affecting foam stability
and structure.
Keywords: rheology, foams.
INTRODUCTION
Aqueous foams are used in the petroleum industry for
enhanced recovery or during drilling operations. As a
drilling fluid, foam is used for drilling depleted reservoirs, in
particular reentry operations, or for drilling in underbalanced
conditions. Among the existing nonconventional drilling
operations, underbalanced drilling offers several benefits
under special conditions. In underbalanced drilling, the
pressure of the drilling fluid is maintained below the
formation pore pressure. It enables to increase penetration
rate, to prevent lost circulation and formation damage and
to minimize differential sticking. Special fluids are used
in order to achieve underbalanced conditions: gas (nitrogen
or air), mist (dispersion of droplets into gas), aerated mud
or foam.
588
Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5
The number of underbalanced drilling operations should
increase during the next decade, as it allows to recover some
old depleted areas and to access to new ones where the
formation pressure is low.
Foam is adequate for underbalanced drilling thanks to its
low density and its good cuttings transport capacity. It is
necessary to know precisely the evolution of the injected
fluid under bottom hole conditions. Indeed, bottom hole
pressure control is essential in underbalanced drilling. This
pressure is strongly dependent on the kind and velocity of
injected fluids, on reservoir pressure and on temperature.
This is all the more true for horizontal wells. In order
to predict accurately the pressure drops along the well, it
is necessary to have a better understanding of foam
rheological behavior, especially under high pressure and
high temperature conditions (which are bottom hole
conditions). This work is a literature review of different
studies of foam rheology, and tries to point out the
important parameters that control foam behavior before any
experimental investigations.
1 GENERALITIES
Foams are dispersions of a gaseous phase in an aqueous
phase containing surfactants. The foam “quality” is the
volume fraction of gas. This volume fraction is generally
very high and can exceed the “packing concentration” of a
hard spheres suspension. For this kind of foams (called “dry”
foams as opposed to “wet” foams where the suspending
liquid phase volume fraction is high), gas bubbles are no
longer spherical but warp into polyhedral bubbles separated
by thin liquid films. The liquid phase is then principally
contained in these thin films and in the “Plateau borders” at
polyhedral bubbles intersections. This kind of system is
considered as a continuous phase only if the bubbles size is
small compared with the sample size.
If we study foam rheology, many parameters will play an
important role. The bubbles size will have a major impact as
well as the interfacial tension and the gaseous phase volume
fraction. The bubbles size is often omitted as a parameter in
experimental studies; yet this average size is likely to vary
with time by gas diffusion from small bubbles to large
bubbles or by coalescence; with foam production method (for
example when the foam is created from flow of gas and
liquid through a porous media: the bubbles size changing
with the imposed flow rate); or with the shear rate
experienced by foam (its structure is modified by viscosity
measurements). Foam is an unstable system: a foam evolves
by drainage of the liquid phase with time, by gas bubbles
coalescence or by Ostwald ripening. Finally, the gaseous
phase compressibility is an additional difficulty when one
carries out measurements on flowing foams.
The rheological behavior of a foam presents singularities
that one must consider: under a weak shear stress, foam
behaves like a plastic solid, i.e. there is a yield stress and
foam begins to flow when it is exceeded. Under a weak
shear stress, foam can flow in “plug flow” by slipping on a
thin fluid layer at the wall. This second point is very
important: there is a wall slip velocity one must take into
account. With all these parameters, foam rheology is a very
complex problem which is a matter of debate at present
time.
Foam flow and therefore foam rheology analysis have
been the subject of numerous studies since about thirty years.
We can divide these works into three major families:
– Experimental studies of foam rheology carried out with
rheometers and laboratory configurations; numerous
parameters are considered (yield stress, wall slip velocity,
drainage phenomena, etc.), from these measurements, a
rheological equation is proposed.
– Foam rheology measurements in conditions simulating
petroleum industry reality: pressure drops measurements in
large size pipes, comparison with foam flow model (SPE
works). Many of these experiments consider foam flowing
into porous media, since foam is used for enhanced oil
recovery.
– Theoretical studies trying to describe foam rheology from a
microscopic model: bubbles described as polyhedral
bubbles, interfacial viscosity, etc. (Kraynik, Princen).
In this paper, we will focus on experimental studies in
order to determine the important parameters to be considered
for good and reproducible viscosity measurements.
2 MEASUREMENT APPARATUS GEOMETRY
Two publications give a general sight over the numerous
studies made on foam rheology (Heller and Kuntamukkula,
1987); Prud'homme and Khan, 1996). The first impression is
the lack of agreement between the different results and the
frequent non-reproducibility of the measurements. The
numerous parameters playing a non-negligible role we tell
about in the introduction do explain this lack of coherence
between the results: it is difficult to control all these
parameters at the same time. Figure 1 gathers experimental
measurements from three authors on the same graph and
shows the disparity of the results for a same foam quality.
The first point is to determine the most adequate
experimental system in order to measure foam viscosity. The
characteristic size of the measurements apparatus must be
widely larger than the average size of the gas bubbles in the
foam, this average size being able to vary from a few
micrometers to a few millimeters depending on pressure or
shear rate conditions. A several millimeters gap at least is
necessary in a coaxial geometry, which is attainable with
difficulty. Moreover, in such a geometry, the volume of foam
B Herzhaft / Rheology of Aqueous Foams: a Literature Review of some Experimental Works
90 quality foam
80 quality foam
1000
10 000
b
100
Effective viscosity (cP)
Effective viscosity (cP)
a
10
1
0,1
589
Mitchell, 1971
Sanghani and Ikoku, 1983
Beyer et al., 1972
1
10
100
Shear rate (s-1)
1000
1000
100
10
1
10 000
Mitchell, 1971
Sanghani and Ikoku, 1983
Beyer et al., 1972
1
10
100
Shear rate (s-1)
1000
10 000
95 quality foam
10 000
Effective viscosity (cP)
c
1000
100
10
1
Mitchell, 1971
Sanghani and Ikoku, 1983
Beyer et al., 1972
1
10
100
Shear rate (s-1)
1000
10 000
Figure 1
Effective viscosity of foam as a function of shear rate and foam quality. From Underbalanced Drilling Manual, Gas Research Institute, 1997.
sample is constant, the drainage phenomena and the foam
compressibility will affect its stability during measurements.
Khan, Schnepper and Armstrong (1988) have studied foam
rheology with a parallel plate rheometer where foam is
placed between two concentric discs. The gap is 2.4 mm
high and authors have made sure that foam is stable during
measurement time: with an imposed shear rate, they make
sure that the torque measured is stable, a decreasing torque
after some times shows degradation of foam. They also
measured bubbles average size by an optical method before
introducing foam into the gap, interfacial tension of liquid
films and yield stress by a “stress relaxation” method. Khan
et al. solved the slip velocity problem by an original mean:
they covered rheometer walls with sand paper; measurements with different gaps showed similar results, which
proved that there was no more wall slip velocity. Their
results show that foam behaves like a Bingham plastic,
effective viscosity varies between 60 and 3000 Pa·s
and increases with foam quality; unfortunately, their
measurements are limited to weak shear rates (from 10-2
to 10-1 s-1), the geometry used doesn't allow larger shear
rates (Fig. 2).
Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5
590
104
104
φ
0.97
0.95
0.92
a
φ
0.97
0.95
0.92
b
103
η (Pa • s)
η (Pa • s)
103
102
101
10-3
102
10-2
10-1
101
100
0
20
40
Tyx (Pa)
γ (s-1)
Foam viscosity as a function of shear rate for three different gas
volume fractions.
60
80
Viscosity as a function of shear stress for three different gas
volume fractions. The vertical asymptotes indicate the yield
stress of the foam.
Figure 2
Effective viscosity of foam as a function of shear rate and stress. From Khan, Schnepper and Armstrong (1988).
It seems that an appropriate geometry in order to study
foam rheology, especially under pressure and high temperature, should be flow in a capillary pipe.
This method consists in measuring the pressure drop
generated by the flow of foam in a capillary pipe versus flow
rate, allowing us to know the shear stress at the wall versus
shear rate. Foam flows continuously, there is yet no more
problems related to a finite volume of foam used for
measurements.
1 to 5 cm. Measurements are performed under weak shear
rate (0 to 6.2 s-1) in steel or acrylic pipes. Results show that
foam flows almost solely by wall slip (Fig. 3) for acrylic
pipes, whereas there is no slip velocity for flow in steel pipes.
8.0
Tw = 70
dyn/cm2
60
6.0
3 WALL SLIP VELOCITY
On the other hand, wall slip problem is still present, and has
been quoted by numerous authors (David and Marsden,
1969; Raza and Marsden, 1967; Beyer et al., 1972; Patton et
al., 1983; Hirasaki and Lawson, 1983; Enzendorfer et al.,
1995). Experiments made on pipes having different
diameters give different results for the same imposed ∆P,
i.e. for the same wall shear stress, which proves the
existence of wall slip velocity. There are methods to
compute wall slip velocity values from these measurements:
Mooney formalism (1931), assuming wall slip velocity is
directly proportional to wall shear stress, has been used by
these authors and gives no satisfactory results. OldroydJastrzebski method (1967), where wall slip velocity is
assumed to be proportional to wall shear stress and inversely
proportional to pipe's diameter, has been used by
Enzendorfer et al. (1995) and gives good results.
Thondavadi and Lemlich (1988) have studied foam flow in
3 m long horizontal pipes having diameters varying from
Qt/πR3(s-1)
50
4.0
40
30
2.0
20
0
0
0.3
0.6
1.2
0.9
1/R (cm-1)
1.5
1.8
Figure 3
Apparent shear rate as a function of the inverse of acrylic
pipe radius for various wall shear stresses. From Thondavadi
and Lemlich (1985).
Harris and Reidenbach (1987) have studied foam rheology
in a recirculating loop made of steel pipes (3 m long and
7.75 mm diameter) and observe no wall slip velocity. The
B Herzhaft / Rheology of Aqueous Foams: a Literature Review of some Experimental Works
nature of the pipe used, especially the state of surface, will
influence results, which is coherent with Khan et al. results in
coaxial geometry. This wall slip velocity is due to the
existence of a thin liquid film at the walls. Some authors have
tried to evaluate this thin film thickness and the wall slip
velocity variation with this thickness (Enzendorfer et al.,
1995; Thondavadi and Lemlich, 1985). According to Calvert
and Nezhati (1986 and 1987), there is a correlation between
this thin liquid film thickness and foam quality, the slip layer
thickness being larger for smaller qualities; but as a first
approximation, they find this film thickness to be
independent of average bubble size.
591
4 YIELD STRESS
The existence of a “yield stress” is much debated because it
is not easily measurable, especially in capillary geometry.
Khan et al. (1988) measured a yield stress in coaxial
geometry with the “stress relaxation” method: a shear rate is
imposed then stopped, the torque value falls then to a nonzero value which is identified as the yield stress value. This
yield stress value increased with foam quality and varies
between 10 and 20 Pa. Numerous yield stress measurements
methods do exist, among these are the use of a “vane device”
or the stress relaxation method (Fig. 4).
b
50
a
17 000
2 Rv
Strain (%)
Bottom view
25
15 000
H
16 000
14 000 (dyn/cm)
0
0
Vane device for the measurement of yeld stresses.
The vane which is attached to the torque transducer on a stress
Angular deflection vs. time for an emulsion measured using a
vane geometry. A serie of increasing shear stresses in posed on
the emulsion. Below the yield stress the emulsion deforms and
then comes to rest; above the yield stress the cup continues to
rotate as the material shears.
Torque reading
c
Rotation rate
Vane device for the measurement of yeld stresses.
The vane, which is attached to the torque transducer on a stress
rheometer, is immersed in a larger cup containing the fluid with a
yield stress.
0
Time
300
150
Time (s)
Yield stress ζ
0
Time
Stress relaxation technique used to measure yield stress in foam. The figure at left shows the rotation rate
applied to the upper disk of the parallel-plate geometry, and the figure at right shows the observed torque
response. The residual stress after a steady shear flow corresponds to the yield stress.
Figure 4
Techniques to measure yield stress. From Prud'homme and Khan (1996).
Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5
592
When using capillary geometry, yield stress value is
measured by extrapolating zero shear rate values. Some
authors (Barnes and Walters, 1985) do consider that yield
stress notion is erroneous and only represents “what can't be
measured”. Kraynik (1988) showed experimentally the
presence of a yield stress in foam by observing near wall
bubbles behavior in a transparent pipe. This “yield stress”
notion is justified by theoretical arguments: it represents the
amount of energy which is necessary to pass from a stable
bubbles network to an other (Khan et al., 1988) (Fig. 5). When
shear stress is less than yield stress and when wall slip velocity
is present, foam does flow entirely by “plug flow”. Beyer et al.
(1972), Kraynik (1988) and Thondavadi and Lemlich (1985)
made experimental observations of this plug flow.
γ
+
1
2
+
1
3
+
+
2
3
Unstrained
a
b
+
+
1
2
+
+
Reformed
d
Strained to
stability limit
γ0 = 2/ 3
c
Sequence showing the deformation of monodisperse, twodimensional, hexagonal foam cell in shear. The gas volume
fraction is unity in this illustration. Increasing strain is applied
moving through the sequence in a clockwise direction from a to d.
The initial cell orientation shown here is θ = 0 relative to the
direction of shear. In c the critical strain is reached where the cells
become unstable, because side "3" has decreased to zero length.
Figure 5
Yield stress of a two dimensions hexagonal foam. From
Khan, Schnepper and Armstrong (1988).
5 FOAM TEXTURE AND STABILITY
Foam texture is given by gas bubbles size distribution. Bubbles
size may vary from a few microns for a fine foam to a few
millimeters for a coarse foam. As we say in introduction,
capillary pipe diameter must be widely greater than bubbles
size — a ratio of ten seems to be a minimum (Prud'homme and
Khan, 1996) — if we want to measure rheological properties of
foam as a continuous medium. Numerous experimental works
measure foam rheology in capillary pipes of very little
dimension (less than one millimeter in diameter). This kind of
analysis is more suitable to the study of foam behavior in a
porous medium, the porous medium being represented by a
group of little capillary pipes.
Foam is an unstable system: it can deteriorate by drainage
of the liquid phase due to gravity. Liquid accumulates then at
the bottom of the sample and foam can't be any longer
considered as an homogeneous system (Princen, 1990).
Bubbles size distribution of a polydisperse foam will also
vary with time: pressure inside little bubbles is greater than
pressure inside large bubbles, gas will then diffuse from
small bubbles to large bubbles, changing distribution of sizes.
Foam must be stable for at least measurements time. Bubbles
size measurements and rheology measurements at the same
time seems to be a necessity to numerous authors
(Prud'homme and Khan, 1996; Calvert and Nezhati, 1986
and 1987; Harris, 1983 with Reidenbach, 1985, 1987, 1990,
1994, 1996) carried out a series of experimentations on
foams and measured textures by an optical method or with a
multichannel particles detector. Foam flows in a recirculating
loop made of steel pipes, pressure drop over a certain
distance being measured and rheological values being
computed from these measurements. A constant shear rate
(i.e., a constant flow rate) is imposed until foam texture is
equilibrated, which can last thirty minutes depending on the
kind of foam being studied. Foam texture is then dependent
on the imposed shear rate (Fig. 6). When foam is
equilibrated, Harris records pressure drops versus different
flow rates, assuming values are measured rapidly enough so
that foam texture does not have time to vary.
6 FOAM PRODUCTION
The foam production method must be well characterized. This
production method has an influence on foam texture and
quality and therefore on foam rheology; it is necessary to form
a stable and well characterized foam in order to have
reproducible measures. The most common foam creation
method consists in injecting surfactant solution and gas
through a porous medium. By varying gas and fluid flow rate
ratio, it is possible to change foam quality. This porous
medium is often constituted of sand (Burley and Shakarin,
1992; Enzendorfer et al., 1995; Raza and Marsden, 1967) or
of stacked glass beads (Patton et al., 1983; David and
Marsden, 1969); Harris (1983, 1985, 1987, 1990, 1994, 1996)
produces foam by circulating first the fluid in the recirculating
loop, and then injecting gas into fluid through a small opening
(the fluid in excess is allowed to escape through a
backpressure regulator). When using a mixer (Hanselmann
and Windhab, 1996), foam quality can’t be controlled.
By varying gas and fluid injection rate into porous media,
foam texture may be modified. Measurements made at
different shear rates (i.e., made at different flow rates) are
carried out on foams having different textures. Patton et al.
solved this problem by imposing a constant flow rate at the
porous medium entrance: the foam texture is then stable; a
bypass pipe at the capillary entrance (i.e., at the porous
B Herzhaft / Rheology of Aqueous Foams: a Literature Review of some Experimental Works
593
2.8
a
2.0
1
D∆P/4L (lb • ft2)
Volume frequency
b
1500 s
1100 s
500 s
250 s
90 s
2.4
1.6
1.2
0.1
1500 s
1100 s
500 s
250 s
90 s
0.8
0.4
0.0
0
400
800
1200
1600
2000
Equivalent spherical diameter (µm)
2400
0.01
10
100
1000
8V/D (s)
Frequency plots show smaller bubbles and narrower distributions
from higher shear rates for equilibrated 70 quality foams.
Rheograms show higher viscosity for foams equilibrated at low
shear rate.
2.8
Quality
c
50
60
70
80
90
2.0
1.6
Quality
d
1
D∆P/4L (lb • ft2)
Volume frequency
2.4
1.2
0.8
90
80
70
60
50
0.1
0.4
0.0
0
400
800
1200
1600
2000
Equivalent spherical diameter
2400
0.01
10
100
1000
8V/D (s)
Texture plot shows 90 quality foam skewed to large bubble size.
Rheograms of 50-90 quality foam.
2.8
2.0%
1.0%
0.5%
e
2.0
f
1
D∆P/4L (lb • ft2)
Volume frequency
2.4
1.6
1.2
2.0%
1.0%
0.5%
0.1
0.8
0.4
0.0
0
400
800
1200
1600
2000
Equivalent spherical diameter
2400
Higher surfactant concentration gives more uniform bubble size
for 90 quality foams.
Figure 6
Experimental results of Harris (1985): texture and foam rheology.
0.01
10
100
8V/D (s)
1000
Higher surfactant concentration gives higher viscosity for 90
quality foams.
Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5
594
medium exit) allows the flow rate to vary (and therefore the
shear rate) in the capillary pipe without changing foam
texture (Fig. 7). Using this technique, large quantities of
foams are produced and then must be treated. With the
recirculating loop used by Harris, this disadvantage is
eliminated, measurements are carried out on equilibrated
foams, flow rate can vary without affecting foam texture if
measurements are made quickly enough.
PG
Liquid
Liq. rotameter
PG
PG
Manometer
Regulator
Capillary tubes
Gas
N2
Packed bed
foam generator
Manifold
By-pass
n
τ
 γ˙ 
= K 
 ε
ε
·
where τ is the shear stress, γ is the shear rate and K and n are
constant.
Experimental measurements are made on very large pipes
(where the pressure drop due to the flow is large enough to
induce foam quality variations) and the agreement is correct
for large quality foams (> 70%) (Winkler, Valko and
Economides, 1994); more recently, Enzendorfer et al. (1995)
made measurements on small dimension pipes (D ≅ 1 cm)
under various pressures (up to 70 bar) and for foams of
different qualities. They validate their results with Valko and
Economides model and obtain good agreement for large
shear rates (Fig. 8). Hanselmann and Windhab (1996) studied
rheology of foams flowing into a pipe under moderate static
pressure conditions (up to 3 bar), taking into account foam
compressibility. They consider a pressure drop and compute
the corresponding volume variations, using the equation of
state of perfect gases for the gas. The volume variations are
then measured experimentally in an expansion tube, and
a good agreement is found with predicted variations.
Variations of viscosity with pressure were studied: effective
viscosity of foam decreases with increasing static pressure.
Figure 7
100
7 FOAM COMPRESSIBILITY
Foam is made principally of compressible gas: a pressure
change will have an influence on the sample volume. If the
pressure drop in the capillary pipe is large, the foam
compressibility must be taken into account. In underbalanced
drilling, pressure drops along the wells are very large, foam
compressibility will then play a major role. Lord (1981)
presented a review of the equations of state and of the
measurement methods proposed for compressible foams. The
pressure drop due to flow through a pipe and the hydrostatic
pressure do interact by means of density variation. Iterative
calculation methods to compute pressure drop in a capillary
pipe were developed by Beyer et al. (1972), Krug and
Mitchell (1972) and Blauer and Kohlaas (1974). Valko and
Economides (1992) developed an analytic method to
compute pressure in a well. They introduce the notion of
“specific volume expansion ratio” defined as the ratio of
foam and liquid (in the foam) densities, compressibility is
taken into account by the mean of a viriel like equation of
state (restricted to the first terms). Foam quality and then
“specific volume expansion ratio” ε vary with pressure. They
consider that foam follows a rheological power law for every
quality or pressure:
τW/ε (Pa)
Experimental system, Patton et al. (1983).
Γ = 48 to 50%
(ρ = 517 to 539 kg/m3)
Γ = 57 to 60%
(ρ = 421 to 447 kg/m3)
Γ = 67 to 70%
(ρ = 324 to 355 kg/m3)
10
1
1
10
8u/(Dε) [s-1]
100
Figure 8
Rheological measurements presented with Valko and
Economides model. From Enzendorfer et al. (1995).
Harris (1983, 1985, 1987, 1990, 1994, 1996) also studied
foam rheology under pressure (69 bar) and high temperature
(149°C) in a recirculating flow loop. However, there are very
few results on foam behavior under high static pressure. The
stability of a static foam under high pressure has been studied
by Rand and Kraynik (1983). They measure drainage
velocity of an aqueous foam formed under pressure (1 to
20 bar) in an autoclave. Their results show that drainage time
B Herzhaft / Rheology of Aqueous Foams: a Literature Review of some Experimental Works
does increase a lot with pressure, the explication for this
enhanced stability is a diminution of the bubble size
— measured with microphotography — (Fig. 9).
Liquid drainage times (min)
120
to deduce the rheological law for a certain quality from the
one for a given quality. They introduce the notion of
“specific volume expansion ratio” (ratio of foam and liquid
densities) noted ε. The rheological equation is then (see
Section 7):
n
τ
 γ˙ 
= K 
 ε
ε
a
80
40
50%
25%
0
0
10
20
Pressure (atmospheres)
Effect of autoclave pressure on the aqueous foam liquid drainage
times.
b
20.0 atm
Viscosity variations with pressure have been studied by
Harris (1985) and by Hanselmann and Windhab (1996). The
apparent viscosity of foam decreases with increasing static
pressure, especially at low shear rates. Harris shows
concurrently that bubbles mean size decreases with pressure
but that the distribution of size is broader for low pressure
conditions.
Harris studied also the influence of temperature. When
temperature increases, it is possible to work with a constant
volume (and therefore to let the pressure increase), or to work
with a constant pressure, allowing the volume to vary. Harris
chose the first option and had the temperature varying up to
149 °C. The apparent viscosity at a given shear rate decreases
with increasing temperature, this diminution being stronger
for low quality foams than for high quality foams.
Finally, most of aqueous foams are adequately described
as non-newtonian fluids following a power law with or
without yield stress, and the viscosity of which decreases
with increasing pressure (or temperature) and decreasing
quality.
100 microns
1.0 atm
595
72.0 atm
Figure 9
Effect of pressure on drainage time and on bubbles size.
From Rand and Kraynik (1983).
8 RHEOLOGICAL EQUATIONS
Different rheological equations are proposed to describe
experimental results. Foam is described as a non-newtonian
fluid:
– pseudoplastic power-law (Thondavadi and Lemlich, 1988;
Raza and Marsden, 1967; David and Marsden, 1969;
Patton et al., 1983; Sanghani and Ikoku, 1983; Enzendorfer
et al., 1995);
– Bingham plastic (Khan et al., 1988);
– Herschel-Bulkley (Calvert and Nezhati, 1986; Burley and
Shakarin, 1992; Harris, 1985).
A power law seems therefore appropriate to describe foam
behavior. All authors agree that foam apparent viscosity for a
given shear rate increases with foam quality, Valko and
Economides (1992) propose an analytical relation that allows
CONCLUSION
Regarding all the studies already done on foam rheology,
a conclusion is that it is necessary to control different
parameters when viscosity measurements are carried out.
Texture (bubbles size distribution) should be characterized
concurrently to rheology measurements. Viscosity measurements at different shear rates, i.e. different flow rates,
must be carried out on identical foams (in particular of same
texture) which are stable during measurement time. In
addition, wall slip velocity and foam yield stress must be
taken into account; in the same way, gas compressibility
should play a role in pressure drops calculation. An
experimental investigation of foam rheology should
consequently present the following characteristics: a foam
formulation stable during measurement time, a method to
measure continuously foam texture (bubbles mean size), a
rheological measurements system which allows to work on
stable and equilibrated foams, and special care about wall
slip velocity and yield stress measurements. Finally,
experimental investigations must pay attention to the
importance of the relation between structure (i.e. foam
texture) and rheology of foam.
596
Oil & Gas Science and Technology – Rev. IFP, Vol. 54 (1999), No. 5
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Final manuscript received in July 1999